# A Join-Based Hybrid Parameter for Constraint Satisfaction

## TL;DR

This paper introduces joinwidth, a new hybrid complexity parameter for CSPs that combines structural and relational aspects, enabling polynomial and fixed-parameter algorithms for broader classes of instances.

## Contribution

The paper defines joinwidth, demonstrating it generalizes existing parameters like hypertree width and fractional hypertree width, unifying various tractable classes of CSPs.

## Key findings

- Joinwidth captures larger tractable classes than structural parameters.
- Bounded joinwidth classes include those with bounded fractional hypertree width.
- Joinwidth unifies multiple hybrid restrictions, explaining their tractability.

## Abstract

We propose joinwidth, a new complexity parameter for the Constraint Satisfaction Problem (CSP). The definition of joinwidth is based on the arrangement of basic operations on relations (joins, projections, and pruning), which inherently reflects the steps required to solve the instance. We use joinwidth to obtain polynomial-time algorithms (if a corresponding decomposition is provided in the input) as well as fixed-parameter algorithms (if no such decomposition is provided) for solving the CSP. Joinwidth is a hybrid parameter, as it takes both the graphical structure as well as the constraint relations that appear in the instance into account. It has, therefore, the potential to capture larger classes of tractable instances than purely structural parameters like hypertree width and the more general fractional hypertree width (fhtw). Indeed, we show that any class of instances of bounded fhtw also has bounded joinwidth, and that there exist classes of instances of bounded joinwidth and unbounded fhtw, so bounded joinwidth properly generalizes bounded fhtw. We further show that bounded joinwidth also properly generalizes several other known hybrid restrictions, such as fhtw with degree constraints and functional dependencies. In this sense, bounded joinwidth can be seen as a unifying principle that explains the tractability of several seemingly unrelated classes of CSP instances.